Dedekind domain

Ordinary integers enjoy unique factorization into primes, and so do polynomials over a field. But the rings of integers of number fields often do not — the famous failure 6 = 2 * 3 = (1 + sqrt(-5))(1 - sqrt(-5)) in Z[sqrt(-5)] shows numbers splitting two genuinely different ways. Dedekind's great repair was to factor ideals rather than numbers. A Dedekind domain is precisely the kind of ring where that repair always works.

A Dedekind domain is an integral domain R that is Noetherian, integrally closed in its field of fractions, and of Krull dimension one (every nonzero prime ideal is maximal). These three conditions together are equivalent to the single striking property: every nonzero proper ideal factors uniquely as a product of prime ideals. Equivalently, every nonzero fractional ideal is invertible, so the fractional ideals form a group.

The motivating examples are the rings of integers O_K of number fields, and more geometrically the coordinate rings of smooth affine curves. A Dedekind domain is a unique factorization domain if and only if it is a principal ideal domain, which happens exactly when its class group is trivial; the class group is the obstruction.